Mutual impedance of N > 2 dipoles (aka fan dipoles)

[rhb]

Does anyone know of a mathematical treatment of the mutual impedance of a fan dipole?  Kraus discusses the case of 2 dipoles.  It should be possible to develop the equations for the case N > 2, but it looks like a good bit of work.  So if someone has done it, I'd prefer just to write software to solve it.

In particular, I have in mind solving the case of a stagger tuned dipole covering 2-50 MHz in the form of a bundle of wires cut to the correct lengths.  Coupling to ground and nearby structures, trees, etc will of course, alter the required lengths. My hope is that by measuring the impedance with a VNA I can determine how much to trim each conductor so that I can raise the antenna into position, measure it, take it down, trim it and put it back up and be done.  Probably not likely to happen, but consider it a "stretch goal". :-)

I found references to the "new Stanford Research Institute method for designing fan dipoles", but cannot find an SRI  paper describing it.

[PhilipPeake]

Honestly, I hate to think what the mathematical relationship looks like.

Ignoring ground, feeder etc effects, my gut feel based upon actually tweaking one is that you would end up with N simultaneous differential equations (N being the number of elements).

[rhb]

It's probably far worse than that.   But I haven't really bothered to analyze the computational complexity in detail yet. 

I plan to run some experiments using ribbon cable over an aluminum ground and my VNWA 3E over 200-800 MHz.  I'm hoping that it can be treated as a series of transmission lines with different properties in each segment as one moves from the center feed point  towards the ends.  That implies reflections in each antenna where each shorter antenna terminates.  That's a very familiar problem in the elastic wave case, but I've never really done any detailed work on an EM case.  The Z transform representations are the same, but the coupling is different.

I was able to get a copy of ECOM-3333 "Multiband HF Antenna" by Robert E. Whitman courtesy of the CECOM archivist who is looking for other Army reports on antennas.  I plan to search through all the IEEE stuff the next time I go to Little Rock.  The UALR library is the only affordable access I have.

I did a good bit of searching for literature using google scholar, but didn't find much that looked promising.  The thing I found odd is it seems to me that whether you have a fan dipole or multiple dipoles, you still have the same coupling issue. 

I suspect that the coupling problem has been masked by the effect of terrain near the antennae.  And if the conductors are not parallel then you get gradients in the impedance and it gets really messy.  If you're going to have to tune it anyway, there's not a lot of incentive to look for a theoretical solution.  Unless you're slightly crazy, a bit bored and putting up your first serious antenna ;-)

[T3sl4co1l]

Think in terms of a conical dipole or bowtie instead.

At a glance, the drawings I see for "fan dipole" look a bit superstitious: stacking up a bunch of conductors in the hopes that they each do their own band, nevermind the (necessary) interaction between them in space (they are quite close, much much less than a wavelength apart), or at the feedpoint.

The equivalent circuit of such an antenna could quite reasonably be, a bunch of twin-lead transmission lines, of different lengths, in parallel.  Probably the resulting 1/4 wave stubs act to tune the overall system, but I'd have to guess at a model and simulate it, to decide if those are just going to notch the response, or actually help bandwidth.

The traditional wideband connection is to space the elements a quarter wavelength along an axis (which becomes the directional axis, so you get that for free), and flip polarity every element (because of phasing and resonance).  In other words, a log periodic.

Or on the same origin (no displacement): rotate the elements relative to each other, thus effectively making a conical or bowtie dipole out of wireframe.  It would seem undesirable for the elements to be different lengths, if that's the case (but you see that, too).  Again, I'd simulate it and see.

Tim

[rhb]

A discone is the continuum case.  They've been done at HF, but are unwieldy.

Robert Whitman's work is actually rather better than what you see in the ham literature. It's a classic engineering compromise to cover 2-4 bands.  The antenna has better gain with the high bands on top.  Also with 3-4 dipoles, the bandwidth of the higher frequency dipoles is significantly reduced.  He also advises a greater than 28% difference in frequency and 5' or 0.03 the length of the lowest frequency element separation at the terminal end of the high frequency dipoles.

Based on what I've learned so far, I'm going to put up either an 80 or 40 m 1/2 wave dipole and study the problem.  As I have a VNWA 3E waiting to go to work, I plan to see if I can solve for the ground reflection by measuring a dipole.  One of the things that makes it so interesting is there is so much mathematics I know which is relevant to the problem.   But it's also different, so it doesn't feel like doing the same thing again.

I suspect that the angular separation of the elements results in an impedance gradient which helps reduce the magnitude of the reflection at the termination.

After spending  a few hours reading the ARRL antenna book and Kraus, I came away with a sense that it is part science and part superstition with a lot of the latter in the ham community.

[vk6zgo]

Mostly, the " superstitious" approach is a good approximation where multiple bands need to be covered, as in Amateur Radio use.
The impedance of the resonant antenna largely overrides that of the others, so it is pretty much OK to ignore them initially, then "tweak" on test.

If you are trying to get wide bandwidth in one band, that is a very different can of worms.

[ocw]

Attached is how EZNEC/NEC2 calculates the SWR of a fan dipole up 10M and the same dipole 62.5M (1/2 wavelength at 2.4 MHz) high.  This includes three dipole elements resonant at 2.4, 3.6 and 5.2 MHz.  I was relying on the odd harmonic resonances to cover the higher frequencies.  The third harmonics were reasonable, but those of higher multiples, while still resonant, had too high of a resistive component.

At a moderate antenna height of 10M the antenna's low frequency resistance was pretty low causing SWR problems at both frequency extremes.

The final SWR chart shows what happens when a 27.4 MHz dipole is added to the 10M high fan dipole.  The other resonances around that frequency also improve while the upper frequency extreme gets worse.

 

[rhb]

The amateur radio bands are generally well separated in frequency which meets the 28% or greater separation. But Whitman's results show that far more separation at the ends is needed than the SRI report states and the typical ham implementation uses.  Unfortunately the scanned PDF is too large to post as an attachment.

I'm not convinced that there is a significant difference between widely separated and closely spaced resonances.  In fact, I could go on at length as to why there is no difference between the two cases that is not readily predictable from basic theory.

The ARRL antenna handbook discusses stagger tuning two dipoles at 80 m to broaden the response.  The more I look at it, the more it looks like a problem in reflection seismology that was solved many years ago.  As I am not currently licensed, putting up an 80 m 1/2 dipole as a receive antenna makes good sense and I can then attack the problem with the VNA to broaden the resonance with a 2nd dipole.

[rhb]

@ocw

At present I am completely ignorant of what the modeling codes are doing relative to the theory.  Having maintained  a few million lines of similar seismology  codes, I am all too acutely aware that "actual mileage may vary".  Such codes are extremely difficult to get correct and even then, you are always forced to make approximations.  In fact, in choosing a method it's usually a case of "and what approximation errors  and artifacts will you accept?" 

Wave equation simulations are intrinsically difficult.   I once had my manager come to my office saying he wanted a simulation run.  I asked him what he wanted, rattled off a list of methods, pulled books off my shelf.  It went nowhere.  None of it meant anything to him, until I mentioned a commercial package that implemented the first method I had mentioned.  At which point he said, "Oh, yes, that's what I want."  It's a very good program, but limited to a 2D model with planar layers. So very basic.  I spent many days working on a simialr code in grad school.  But all he knew was the trade name.

Would you be so kind as to post ASCII files that would let me plot the results?  That would make it easier for me to compare them.  Also the model information.  I very much appreciate your taking the time to do the modeling.

[ocw]

I have designed and built a lot of antennas.  For many of them I have confirmed my design of the antenna via field strength readings or reverse field strength readings [activating a remote beacon transmitter and antenna, and using that signal to evaluate the receive strength of that signal on my test antenna to determine its directivity.  It is also used to compare relative receive strength of the test antenna compared to a reference antenna, such as a dipole].

I have found the constructed and tested antenna's performance to closely match the modeling via EZNEC6/NEC2 software.  The software's accuracy depends primarily on how accurate your modeling parameters are.

I tried to design an antenna to meet the originally stated goal:  2 to 50 MHz.  The resonant frequency of one dipole wire in a fan array was not the same as for that same length wire by itself.  I would have picked different parameters for a ham band antenna over the same frequency range. 

I suggest doing what I did--start with individual dipoles for your preferred frequency, combine them and then alter their parameters to try and meet your goals as best that you can.  You will need to decide what sort of compromises you are willing to accept.

[rhb]

Based on Whitman's report it would take about 100 dipoles to span 2-50 MHz at a 4% BW for each.  That's not possible with the SRI/ECOM design.

For a station antenna, I'm going to put up a conventional mono band 1/2 wave dipole to start with and measure it very carefully.  I should be able to solve for the ground properties by comparison to the free space response.  The trees are likely to make that a bit difficult, but I should be able to just lump all the environmental effects into a single term.  Then I'll add a second dipole and attempt to quantify the coupling effects.  That gets me a proper antenna for RX and a bit of familiarity with EM coupling.  Naturally, this will involve lots of modeling using  the NEC software.

I do not have the grasp of EM that I have of elasticity.  So this will not be quick.  I have a *lot* of reading to do. However, it's also my 3rd go round with the wave equation.  I've already done light and elasticity.  While I have excellent texts on EM theory,  I have very little on the subject of modelling. So I expect I shall need $400-500  in books.

From a the standpoint of practical experimentation, ribbon cable over a perfect ground from 200-800 MHz is much simpler and cheaper.  I have a large quantity of ribbon cable I can use for which I no longer have much other use. So if I can find a low cost RF absorption material I should be able to do a lot of experiments in a short period of time in a convenient space.

[ocw]

I have a large quantity of ribbon cable I can use...

Ribbon cable has its wires too closely spaced to perform well as a fan dipole covering multiple frequencies.  It will act more as fatter single wire dipole of its longest length.  And, it won't be strong enough to survive long in most outside environments.

What's the value of a 2-50 MHz antenna?  2-3.5 MHz has no amateur frequencies and only a limited amount of tropical band broadcasting activity.  And, as my modeling shows, a dipole won't perform the best as a low height.  Other frequencies will probably also not be that important for you.  A single wire low frequency dipole will perform reasonably as an "all HF frequency" receive antenna.

[rhb]

Ribbon cable has well defined spacing and dielectric properties.  It's also very easy to separate a wire from the ribbon and place it at any desired position.  I should have thought it obvious from the context that it's only intended for physical modeling experiments, not for actual use as an antenna.  It's just a means for verifying the numerical modeling results are correct.

I've done a lot of work on numerical simulations, devising test cases. tracking down bugs and advising users of the limitations of various programs.  I've never seen a simulation that always produced the correct answer.  In fact, I rather doubt that such exists in *any* domain.  I know it's not possible in elasticity.  Tell me the method and I'll tell you where it fails and why.  Electromagnetic and elastic waves are both described by the same differential equations.  Except that the behavior of a plane wave at an interface is different, *all* the issues inherent in numerical simulations of the wave equation in elasticity apply to electromagnetism.

[rfeecs]

The SRI study may be this:
http://www.dtic.mil/dtic/tr/fulltext/u2/684938.pdf

It has a few pages, section III, describing a Multiband Antenna.

It also points out that it covers 3 bands, each with about +/-2% bandwidth, not the whole continuous spread of frequencies.

[ocw]

The SRI study...

I like both studying and collecting both old and new reference books.  The classic ones are always nice.  Many of them describe innovative ideas which are referred to long after they were published.  However, sometimes this is still done after parts of those books have been proven to be incorrect or an over simplification.

As an example, besides "standards" like "Antennas" by John Kraus (copyright 1950), the "Antenna Engineering Handbook" by Henry Jasik (copyright 1961) and others, I have several books on AM broadcast antennas by Carl E. Smith of Smith Electronics.  One of them is "Theory and Design of Directional Antennas."  While its original version was published in 1949, I have a copy if the Third Edition which was published in 1969.  I remember going to a conference in the late 70's and talking with a representative of Smith Electronics.  I proudly bragged of having that book and others by Smith.  The representative from Smith Electronics said that they didn't look at those old books anymore since many of them were too outdated to refer to anymore.

So, while the old books might be a good place for first ideas, I would use that as a lead into more recent publications better detailing the same subject using tools which weren't around when those classic books were written.

[rhb]

That is the "SRI study".  I spent a ridiculous amount of time trying to locate it because of the way it was cited.

ECOM-3333 by Robert Whitman in 1970 goes well beyond the SRI manual. Most importantly, he found a significant increase in gain if the high band dipoles were above the low band dipoles.  He also found  that the antennae need to be separated by at least 0.03x the length of a leg of the low frequency dipole at the ends.  That's very different from any of the ham versions I've seen.

I downloaded 20 reports on antennae produced by ECOM/CECOM courtesy of the CECOM archivist this morning.  I've not had time to read any yet, but ECOM-3333 convinced  me that there is a lot of high quality science and engineering done by the US Army which has been overlooked.  I suspect primarily because of the difficulty in finding and accessing the reports when it was all paper based.  I'm hoping to find a simple way to make these available online.  I suspect there is a lot more useful work in the CECOM archives.

At this point I'm largely convinced that physical experiments in the 200-1600 MHz range are needed for which I ordered an xaVNA this morning. Kraus presents a coupling  analysis of two dipoles, but it's not clear how to extend that to N > 2 without encountering combinatorial explosion. The ARRL handbook just treats stagger tuning of an 80 m dipole pair.   From what I've read so far, I'm skeptical of any modeling code handling a large number of dipoles correctly.  While I could not access the papers themselves, the abstracts I found with google scholar did not suggest that the basic work required had ever been done.

With regard to old vs new references, I have a simple approach.  I read them all.  And I read all the important citations, which all too often leads to my discovering the citation has nothing to do with the subject for which it was given.  Unfortunately it's a 150+ mile round trip to get IEEE access.  But that will allow me time to study what I have already.

[rfeecs]

Not sure how much you'll get out of IEEE, but annual membership is about $200, 50% off if you're over 62 I think.  Membership in a specific society, like Antennas and Propagation Society is additional $15.  Gives you full online access to related journals:
https://www.ieee.org/membership-catalog/productdetail/showProductDetailPage.html?product=MEMAP003
So it may not be all that expensive to get convenient access.

[rhb]

I used to be a member of IEEE.  But unlike ACM where you can buy unlimited access to the digital library and SEG where it comes with membership, IEEE has literature access charges  for members which is simply unaffordable.

From the IEEE web site:

"IEEE members receive a discounted price of US$13 on single IEEE article purchases made through IEEE Xplore. Articles from partner publishers are US$33 per article."

or if you prefer volume pricing:

"Access all the IEEE content you need to explore ideas and develop better technology with 25 article downloads every calendar month for just US$44.00 per month. Plus, roll over up to 10 unused downloads into the following month for a maximum of 35 downloads."

I generally collect a lot of papers when I'm working on a project.  But it's bursty.  I can easily go over the 25 paper limit in a couple of hours.  But then I might not want more papers for several months.  Either way it's a lot of money.  My normal practice is to get a dozen or so recent papers, read them and get the important references. read those and get the important references.  A lot of papers turn out to be useless, but until you read it, there is no way to know.

Unfortunately, SPE has similar policies on literature access.  It seems to be an engineering society practice.  The scientific societies I've belonged to provided much better literature access to their members.  After the company where I worked for 6 years as a contractor closed their library I was spending around $1500 a year for society memberships to get literature access.

[rfeecs]

I used to be a member of IEEE.  But unlike ACM where you can buy unlimited access to the digital library and SEG where it comes with membership, IEEE has literature access charges  for members which is simply unaffordable.

From the IEEE web site:

"IEEE members receive a discounted price of US$13 on single IEEE article purchases made through IEEE Xplore. Articles from partner publishers are US$33 per article."

or if you prefer volume pricing:

"Access all the IEEE content you need to explore ideas and develop better technology with 25 article downloads every calendar month for just US$44.00 per month. Plus, roll over up to 10 unused downloads into the following month for a maximum of 35 downloads."

That's not my experience.  I'm a member of IEEE and MTT society for example and have free access to all the products included in the MTT society membership:

IEEE Microwave Theory and Techniques Society Membership
Included products
  - Lightwave Technology, Journal of
       Format:Electronic
  - Microwave Magazine, IEEE
       Format:Electronic, Print, Digital
  - Microwave Theory and Techniques Conference Digital Library, IEEE
       Format:Electronic
  - RFIC Virtual Journal, IEEE
       Format:Electronic
  - RFID Virtual Journal, IEEE
       Format:Electronic
  - Terahertz Science and Technology, IEEE Transactions on
       Format:Electronic
  - Microwave and Wireless Components Letters, IEEE
       Format:Electronic
  - Microwave Theory and Techniques, IEEE Transactions on
       Format:Electronic
  - Multiscale and Multiphysics Computational Techniques, IEEE Journal of
       Format:Electronic
  - Electromagnetics, RF and Microwaves in Medicine and Biology, IEEE Journal of
       Format:Electronic

I can download full text pdf of any paper included in the above list that has Format:electronic.  I don't pay any additional charge.  I think those charges only apply if you are downloading papers not included in the societies that you have included in your membership.

[rhb]

For how much per year?  It appears that IEEE pricing is a Byzantine maze.

[rfeecs]

For how much per year?  It appears that IEEE pricing is a Byzantine maze.

I paid
$201 for IEEE membership,
$18 for Electron Devices society membership,
$24 for Microwave Theory and Techniques Society Membership

For total of $243

[ocw]

I decided to try some more antenna modeling, increasing the number of dipoles from three to eight.  I had them cover 2 - 7 MHz with their third harmonic resonances adding up to 20 MHz.  To increase the feed point resistance at the lower frequencies, I modeled the antenna 62.5M high with real earth ground parameters.  The results looked reasonable and came close to my expectations.  The first attachment shows the 50 ohm SWR over that complete range.

The second attachment shows greater resolution of the fundamental frequency bandwidth of the four lower frequency dipoles.  There was the expected reduced bandwidth on all but the lowest frequency dipole.

The third attachment shows the same thing on the four higher frequency dipoles on the top half of that attachment.  I then shortened the lengths of the four longer dipoles to the exact mid-length point of the four higher frequency dipoles.  That should have made them resonant right between the resonant frequencies of the original higher frequency dipoles.  Unfortunately, that did not actually occur.  The lower half of that attachment shows how the resonance changed.  My first frequency spacing appears to be in the ball park of the minimum practical with the way that I had the dipoles placed.

Other dipole placements might have better results, but I'm giving up on trying to make it any better...

[rhb]

What is the layout of the dipoles? I'd be particularly interested in the case where all the dipoles are parallel and in close proximity with the dipoles at integer fractions of the longest dipole wavelength.  That would tell me quite a lot about the modeling code.

[ocw]

What is the layout of the dipoles?

I had all dipoles horizontal.  If they were supported by a structure in the shape of a circle having the diameter of the longest dipole (with a rope fulfilling the remaining length of the shorter dipoles), they would attach to that circle about 1.25M apart from each other.  A larger distance would obviously be safer, but I was trying to design something that would be more practical to actually construct.  The circumference of the part of 62.5M high circle used for dipole support would be 8.75M on each side.  Do you want to build that for me? 

[rhb]

The FAA would never let me put something up that high ;-(  The best I can hope for is perhaps 15-20 m.

I need to look at the antenna modeling codes and find one I can use in a program to search for the optimal parameters and then abuse a computer or two for a few days.

I wonder if anyone has ported an antenna code to the nVidia CUDA environment?

Edit:

There is a very good presentation by Steve Stearns, K6OIK, "Antenna Modeling for Radio Amateurs" from October of 2017.  He compares the results from 6 different numerical modeling codes.  In general, none of them give the same answers for a 1 meter dipole with an L/d of 50 from 1-1500 MHz.

Based on that I ordered, "Antenna Theory" by Balanis and am giving serious thought to " Higher Order Basis Based Integral Equation Solver" by Zhang et al as it comes with a license for the HOBBIES program.

I've been trying to find a copy of the NEC2 source but so far the only copy seems to be at QSL.org and I'm waiting on an account.  The NEC4 source is available for $300 which is stiff, but not excessive.  HOBBIES with the book costs $231.